﻿ Maximum cut

# Maximum cut

The maximum section of a graph is an assignment set of nodes in its two partitions, so that the total weight of the partitions extending between the two edges is a maximum. In contrast to the minimum cut problem is NP -complete.

• 3.1 2- approximation
• 3.2 (2 ) approximation

## Formal notation

The problem is considered of an undirected graph. Optionally in addition to the graph, an evaluation function, that is, the edge weights.

Wanted is a partition so that maximum.

## NP- completeness

### Decision problem

The corresponding decision problem asks for an input and is there a section whose value is greater than?

## Approximation Algorithms

### 2- approximation

The partitioning of the graph is determined by the status of the node ( on / off). It is now trying to maximize the total value of "good" edges; which are by definition all edges between the partitions. A " flip " operation pushes a node from one partition to the other ( turns it on or off). It is achieved by lining flips a local maximum, as long as by random flips are performed as described by the total value is improved (since the total value is limited, and it increases with every operation it termiert actually sometime ). The problem is, however, that the term is only a function of the total weight pseudopolynomiell.

### (2 ) approximation

In order to achieve real polynomial running time, only flips are made that involve a great improvement in weight, more precisely, increase the ratio by weight of at least a factor. Characterized the weight is multiplied in a linear number of puffs, whereby the maximum weight may be achieved in many logarithmic steps; the solution is somewhat inaccurate, since even if there is a small improvement would be possible, it is no longer made.

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